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Periodic orbits to Kaplan-Yorke like differential delay equations with two lags of ratio \((2k-1)/2\)
Advances in Difference Equations volume 2016, Article number: 247 (2016)
Abstract
In this paper, we study the periodic solutions to a type of differential delay equations with two lags of ratio \((2k-1)/2\) in the form
The 4k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system. This is a new type of differential-delay equations compared with all previous researches since the ratio of two lags is not an integer. Three functionals are constructed for a discussion on critical points. An example is given to demonstrate our main results.
1 Introduction
The differential delay equations have useful applications in various fields such as age-structured population growth, control theory, and any models involving responses with nonzero delays [1–5].
Given \(f\in C^{0}(R^{+},R^{-})\) with \(f(-x)=-f(x),xf(x)>0,x\neq0\). Kaplan and Yorke [6] studied the existence of 4-periodic and 6-periodic solutions to the differential delay equations
and
respectively. The method they applied is transforming the two equations into adequate ordinary differential equations by regarding the retarded functions \(x(t-1)\) and \(x(t-2)\) as independent variables. They guessed that the existence of \(2(n+1)\)-periodic solution to the equation
could be studied under the restriction
which was proved by Nussbaum [7] in 1978 by use of a fixed point theorem on cones.
After then, a lot of papers [8–21] discussed the existence and multiplicity of \(2(n+1)\)-periodic solutions to equation (1.3) and its extension
where \(F\in C^{1}(R^{N},R),F(-x)=F(x),F(0)=0\).
Recently, Zhang and Ge [22] studied the multiplicity of 2n-periodic solutions to a type of differential delay equations of the form
and obtained new results.
In this paper, we study the periodic orbits to a type of differential delay equations with two lags of ratio \((2k-1)/2\) in the form
which is different from (1.3) and can be regarded as a new extension of (1.2). The method applied in this paper is the variational approach in the critical point theory [23, 24].
Since the equation
can be changed into the form of equation (1.5) by the transformation
this paper completes the research of the equations in the form
In fact, it follows from
that
for \(\widehat{f}=2f\), which is much the same as equation (1.5).
We suppose that
and there are \(\alpha,\beta\in R\) such that
Let \(F(x)=\int_{0}^{x}f(s)\,ds\). Then \(F(-x)=F(x)\) and \(F(0)=0\). For convenience, we make the following assumptions.
- (S1):
- (S2):
-
there exist \(M>0\) and a function \(r\in C^{0}(R^{+},R^{+})\) satisfying \(r(s)\rightarrow\infty\), \(r(s)\rightarrow0\) as \(s\rightarrow \infty\) such that
$$\biggl|F(x)-\frac{1}{2}\beta x^{2}\biggr|>r\bigl(|x|\bigr)-M, $$ - (\(\mathrm{S}_{3}^{\pm}\)):
-
\(\pm[F(x)-\frac{1}{2}\beta x^{2}]>0, |x|\rightarrow\infty\),
- (\(\mathrm{S}_{4}^{\pm}\)):
-
\(\pm[F(x)-\frac{1}{2}\alpha x^{2}]>0, 0<|x|\ll1\).
In this paper, we need the following lemma.
Let X be a Hilbert space, \(L:X\rightarrow X\) be a linear operator, and \(\Phi:X\rightarrow R\) be a differentiable functional.
Lemma 1.1
([24], Theorem 2.4; [8], Lemma 2.4)
Assume that there are two closed \(s^{1}\)-invariant linear subspaces \(X^{+}\) and \(X^{-}\) and \(r>0\) such that
-
(a)
\(X^{+}\cup X^{-}\) is closed and of finite codimensions in X,
-
(b)
\(\widehat{L}(X^{-})\subset X^{-},\widehat {L}=L+P^{-1}A_{0}\) or \(\widehat{L}=L+P^{-1}A_{\infty} \),
-
(c)
there exists \(c_{0}\in R\) such that
$$\inf_{x\in X^{+}}\Phi(x)\geq c_{0}, $$ -
(d)
there is \(c_{\infty}\in R\) such that \(\Phi(x)\leq c_{\infty}<\Phi(0)=0, \forall x\in X^{-}\cap S_{r}=\{x\in X^{-}:\|x\|=r\}\),
-
(e)
Φ satisfies \((P.S)_{c}\)-condition, \(c_{0}< c< c_{\infty}\). Then Φ has at least \(\frac{1}{2}[\dim(X^{+}\cap X^{-})-\operatorname{co\,dim} _{X}(X^{+}\cup X^{-})]\) generally different critical orbits in \(\Phi ^{-1}([c_{0},c_{\infty}])\) if
$$\bigl[\dim\bigl(X^{+}\cap X^{-}\bigr)-\operatorname{co \,dim}_{X}\bigl(X^{+}\cup X^{-}\bigr) \bigr]>0. $$
Remark 1.1
We may use \((P.S)\)-condition to replace condition (e) in Lemma 1.1 since \((P.S)\)-condition implies that \((P.S)_{c}\)-condition holds for each \(c\in R\).
In order to construct adequate functional whose critical points are the solutions of equation (1.6), we need to distinguish our problem into three cases:
and
Then we construct the corresponding three functionals.
2 Space X, functional Φ, and its differential \(\Phi'\)
2.1 4k-Periodic orbits to equation (1.6) when \(k=3l+2\)
We are concerned at the 4k-periodic solutions to (1.6) and suppose that
We transform (1.6) into
Let
and define \(P:X\rightarrow L^{2}\) by
Then the inverse \(P^{-1}\) of P exists. For \(x\in X\), define
Therefore, \((X,\|\cdot\|)\) is an \(H^{\frac{1}{2}}\) space.
Define the functional \(\Phi:X\rightarrow R\) by
where
Let \(k_{1}= [\frac{2k-2}{3} ]= [\frac{6l+2}{3} ]=2l, k_{2}=k-1=3l+1\), and
We have
For each
define \(\Omega:X\rightarrow X\) by
If \(x_{i}(t)=a_{i}\cos\frac{(2i+1)\pi t}{6l+4}+b_{i}\sin\frac{(2i+1)\pi t}{6l+4}\in X(i), i\in N\), then we have
Obviously, \(L|_{X(i)}:X(i)\rightarrow X(i)\) is invertible.
By the theorem of Mawhin and Willem [25], Theorem 1.4, the functional Φ is differentiable, and its differential is
where \(K(x)=P^{-1}f(x)\). It is easy to prove that \(K:(X,\|x\| ^{2})\rightarrow(X,\|x\|_{2}^{2})\) is compact.
It is easy to see that \(\langle\Omega x,x\rangle=0\). Therefore, from (2.6) we have that if
then
On the other hand,
Therefore, we have
Lemma 2.1
Each critical point of the functional Φ is a \((12l+8)\)-periodic solution of equation (1.6) satisfying (2.1).
Proof
Let x be a critical point of the functional Φ. Then \(x(t)\) satisfies
Consequently,
Subtracting (2.10) from (2.11), we have
that is,
which implies that x is a solution to (1.6). □
2.2 4k-Periodic orbits to equation (1.6) when \(k=3l+3\)
We are concerned at the 4k-periodic solutions to (1.6) and suppose that
We transform (1.6) into
Let
and define \(P:X\rightarrow L^{2}\) by
Then the inverse \(P^{-1}\) of P exists. For \(x\in X\), define
Therefore, \((X,\|\cdot\|)\) is an \(H^{\frac{1}{2}}\) space.
Define the functional \(\Phi:X\rightarrow R\) by
where
Let \(k_{1}= [\frac{2k-2}{3} ]= [\frac{6l+4}{3} ]=2l+1,k_{2}=k-1=3l+2\), and
We have
Define \(\Omega:X\rightarrow X\) by
Then if
then we have
The functional Φ is differentiable, and its differential is
where \(K(x)=P^{-1}f(x)\). The mapping \(K:(X,\|x\|^{2})\rightarrow(X,\| x\|_{2}^{2})\) is compact.
Therefore, from (2.17) it follows that, for each
we have
On the other hand,
Therefore, we have
Lemma 2.2
Each critical point of the functional Φ is a \((12l+12)\)-periodic solution of equation (1.6) satisfying (2.12).
Proof
Let x be a critical point of the functional Φ. Then \(x(t)\) satisfies
Consequently, we have
Subtracting (2.21) from (2.22), we have
that is,
which implies that x is a solution to (1.6). □
2.3 4k-Periodic orbits to equation (1.6) when \(k=3l+4\)
We are concerned at the 4k-periodic solutions to (1.6) and suppose that
We transform (1.6) into
Let
and define \(P:X\rightarrow L^{2}\) by
Then the inverse \(P^{-1}\) of P exists. For \(x\in X\), define
Therefore, \((X,\|\cdot\|)\) is an \(H^{\frac{1}{2}}\) space.
Define the functional \(\Phi:X\rightarrow R\) by
where
Let \(k_{1}= [\frac{2k-2}{3} ]= [\frac{6l+6}{3} ]=2l+2,k_{2}=k-1=3l+3\), and
We have
Define \(\Omega:X\rightarrow X\) by
Then if
then we have
The functional Φ is differentiable, and its differential is
where \(K(x)=P^{-1}f(x)\). The mapping \(K:(X,\|x\|^{2})\rightarrow(X,\| x\|_{2}^{2})\) is compact.
For
we have
On the other hand,
Therefore, we have
Lemma 2.3
Each critical point of the functional Φ is a \((12l+16)\)-periodic solution of equation (1.6) satisfying (2.23).
Proof
Let x be a critical point of the functional Φ. Then \(x(t)\) satisfies
Consequently,
Subtracting (2.32) from (2.33), we have
that is,
which implies that x is a solution to (1.6). □
3 Partition of space X and symbols
In fact, in Sections 2.1, 2.2, and 2.3, we could let
and
and define \(\Omega:X\rightarrow X\) by
Then if \(x_{i}(t)=a_{i}\cos\frac{(2i+1)\pi t}{2k}+b_{i}\sin\frac {(2i+1)\pi t}{2k}\in X(i), i\in N\), then we have
and, for each
we have
On the other hand,
Therefore, we have
Let
On the other hand,
Obviously, \(\dim X_{\infty}^{0}<\infty\) and \(\dim X_{0}^{0}<\infty\).
Lemma 3.1
Under assumptions (S1) and (S2), there is \(\sigma>0\) such that
and
Proof
First, we have that, for \(\beta\geq0\) and \(i\in\{0,1,\ldots ,k_{1}\}\),
where \(m_{0}^{+}(i)=\max \{m\in N:-\frac{(4mk+2i+1)\pi}{4k}\frac{\cos\frac {(2i+1)\pi}{4k}}{ \sin\frac{(2i+1)3\pi}{4k}}+\beta>0 \}\), and
where \(m_{0}^{-}(i)=\min \{m\in N:-\frac{(4mk+2i+1)\pi}{4k}\frac{\cos\frac {(2i+1)\pi}{4k}}{ \sin\frac{(2i+1)3\pi}{4k}}+\beta<0 \}\).
In this case, we may choose
and let \(\sigma^{0}=\min\{\sigma_{0},\sigma_{1},\ldots,\sigma_{k_{1}}\}>0\). Further, for \(\beta\geq0\) and \(i\in\{k_{1}+1,\ldots,k_{2}\}\), we have
where \(m_{1}^{+}(i)=\max \{m\in N:\frac{(4mk+4k-2i-1)\pi}{4k}\frac{\cos \frac{(2i+1)\pi}{4k}}{ \sin\frac{(2i+1)3\pi}{4k}}+\beta>0 \}\), and
where \(m_{1}^{-}(i)=\min \{m\in N:\frac{(4mk+4k-2i-1)\pi}{4k}\frac{\cos \frac{(2i+1)\pi}{4k}}{ \sin\frac{(2i+1)3\pi}{4k}}+\beta<0 \}\).
In this case, we may choose
and let \(\sigma^{1}=\min\{\sigma_{k_{1}+1},\sigma_{k_{1}+2},\ldots ,\sigma_{k_{2}}\}>0\).
Let \(\sigma=\min\{\sigma^{0},\sigma^{1}\}=\min\{\sigma_{1},\sigma _{2},\ldots,\sigma_{k_{2}}\}\). The proof for the case \(\beta<0\) is similar. We omit it. The inequalities in (3.1) are proved. □
Lemma 3.2
Under conditions (S1) and (S2), the functional Φ defined by (2.3) satisfies \((P.S)\)-condition if
for some \(M_{0}>0\) and some function \(r\in C^{0}(R^{+},R^{+})\) that satisfies
Proof
Let \(\Pi,N,Z\) be the orthogonal projections from X onto \(X_{\infty}^{+},X_{\infty}^{-},X_{\infty}^{0}\), respectively. From the second condition in (1.9) it follows that
for some \(M>0\).
Assume that \(\{x_{n}\}\subset X\) is a subsequence such that \(\Phi '(x_{n})\rightarrow0\) and \(\Phi(x_{n})\) is bounded. Let \(w_{n}=\Pi x_{n},y_{n}=Nx_{n},z_{n}=Zx_{n}\). Then we have
From
and (3.4) we have
and then, by (3.1),
which, together with \(\Pi\Phi'(x_{n})\rightarrow0\), implies the boundedness of \(w_{n}\). Similarly, we have the boundedness of \(y_{n}\). At the same time, (S2) yields
Then the boundedness of \(\Phi(x)\) implies that \(\|z_{n}\|_{2}\) is bounded. Consequently, \(\|z_{n}\|\) is bounded since \(X_{\infty}^{0}\) is finite-dimensional. Therefore, \(\|x_{n}\|\) is bounded.
It follows from (2.7) that
From the compactness of operator K and the boundedness of \(x_{n}\) we have that \(K(x_{n})\rightarrow u\). Then
The finite-dimensionality of \(X_{\infty}^{0}\) and the boundedness of \(z_{n}=Zx_{n}\) imply \(z_{n}\rightarrow\varphi\in X_{\infty}^{0}\). Therefore,
which implies \((P.S)\)-condition. □
Lemma 3.3
Under conditions (S1) and (S2), the functional Φ defined by (2.14) and (2.25) satisfies \((P.S)\)-condition if
for some \(M_{0}>0\) and some function \(r\in C^{0}(R^{+},R^{+})\), which satisfies
The proof of Lemma 3.3 is the same as that of Lemma 3.2, and we omit it.
4 Notation and main results of this paper
We first give some notation.
Denote
and
Now we give the main results of this paper.
Theorem 4.1
Suppose that (S1) and (S2) hold. Then equation (1.6) possesses at least
4k-periodic solutions satisfying \(x(t-2k)=-x(t)\), provided that \(n > 0\).
Theorem 4.2
Suppose that (S1), (S2), (\(\mathrm{S}_{3}^{+}\)), and (\(\mathrm{S}_{4}^{-}\)) hold. Then equation (1.6) possesses at least
4k-periodic solutions satisfying \(x(t-2k)=-x(t)\), provided that \(n > 0\).
Theorem 4.3
Suppose that (S1), (S2), (\(\mathrm{S}_{3}^{-}\)), and (\(\mathrm{S}_{4}^{+}\)) hold. Then equation (1.6) possesses at least
4k-periodic solutions satisfying \(x(t-2k)=-x(t)\), provided that \(n > 0\).
5 Proof of main results of this paper
Proof of Theorem 4.1
Suppose without loss of generality that
Let \(X^{+}=X_{\infty}^{+}\) and \(X^{-}=X_{0}^{-}\). Then
Obviously,
which implies that condition (a) in Lemma 1.1 holds. Let \(A_{\infty}=\beta\). Then condition (b) in Lemma 1.1 holds since for each \(j\in N\), we have that \(x\in X(j)\) yields \((L+P^{-1}\beta)x\in X(j)\).
At the same time, Lemma 3.2 gives the \((P.S)\)-condition.
Now it suffices to show that conditions (c) and (d) in Lemma 1.1 hold under assumptions (S1) and (S2).
In fact, condition (S1) implies that on \(X^{-}\) we have \(\Phi (x)>0\) if \(0<\|x\|\ll1\), that is, there are \(r>0\) and \(c_{\infty}<0\) such that
On the other hand, we have shown in Lemma 3.1 that there is \(\sigma>0\) such that \(\langle(L+P^{-1})x,x \rangle>\sigma\|x\|^{2},x\in X_{\infty}^{+}\). On the other hand, \(|F(x)-\frac{1}{2}\beta x^{2}|<\frac{1}{4}\sigma|x|^{2}+M_{1},x\in R\), for some \(M_{1}>0\).
Then
if \(x\in X^{+}\). Clearly, there is \(c_{0}< c_{\infty}\) such that \(\Phi (x)\geq c_{0},x\in X^{+}\).
Our last task is to compute the value of
By computation we get that, for each \(i\in\{0,1,\ldots,k_{2}\}\),
and
Therefore,
if \(i\in\{0,1,\ldots,k_{1}\}\) and \(m\geq0\) is large enough,
if \(i\in\{k_{1}+1,\ldots,k_{2}\}\) and \(m\geq0\) is large enough, which means that there is \(M>0\) such that \(\dim(X_{\infty}^{+}(j)\cap X_{0}^{-}(j))-\operatorname{co\,dim}_{X}(X_{\infty }^{+}(j)+X_{0}^{-}(j))=0, j>M\), from which it follows that
Then we have
and
Therefore,
Theorem 4.1 is proved. □
Proof of Theorem 4.2 and Theorem 4.3
Since the proof for the two theorems is almost the same, we prove only Theorem 4.2.
Let \(X^{+}=X_{\infty}^{+}+X_{\infty}^{0},X^{-}=X_{-}^{0}+X_{0}^{0}\). Then as in the proof of Theorem 4.1, we check conditions (a), (b), (c), (d), and (e). In the present case, we may suppose that (5.1) still holds for some \(M>0\). Let \(X_{\infty}^{0}(i)=X_{\infty}^{0}\cap X(i),X_{0}^{0}(i)=X_{0}^{0}\cap X(i)\). Then
Our proof is completed. □
6 An example
Example
Suppose that \(f\in C^{0}(R,R)\) is such that
We are to discuss the multiplicity of 12-periodic solutions of the equation
In this case, \(k=3,k_{1}=1,k_{2}=2,\alpha=-\frac{7\pi}{2},\beta=\frac {5\pi}{2}\). This yields that
Applying Theorem 4.2, we conclude that equation (6.1) possesses at least 27 different 12-periodic orbits satisfying \(x(t-6)=-x(t)\) since f satisfies (S1), (S2), (\(\mathrm{S}_{3}^{+}\)), and (\(\mathrm{S}_{4}^{-}\)).
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Acknowledgements
Sponsored by the Scientific Research Project of Beijing Municipal Commission of Education (No. KM201511232018) and the National Natural Science Foundation of China (11471146).
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Li, L., Xue, C. & Ge, W. Periodic orbits to Kaplan-Yorke like differential delay equations with two lags of ratio \((2k-1)/2\) . Adv Differ Equ 2016, 247 (2016). https://doi.org/10.1186/s13662-016-0967-3
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DOI: https://doi.org/10.1186/s13662-016-0967-3
MSC
- 34B10
- 34B15
Keywords
- differential delay equation
- periodic solutions
- critical point theory
- variational method